to the integral because, for « = 27/T, the exponential term is 
unity. Let these spectral components travel in the positive x 
direction. Under these conditions, the variation of 7 with x 
can be expressed, and equation (4.4) follows from equation (4.3). 
Equation (4.4) reduces to equation (4.3) if x is set equal 
to zero. In addition, a correct spectral wave length has been 
assigned to each spectral frequency, - Equation (2.25) applies 
where 6 is equal to -7r/2 and 9 is zero. 
In the first term of equation (4.4), the limits of integration 
can be changed from zero through infinity to minus infinity through 
zero by the relations given in equation (4.5), and finally 7 (x,t) 
can be expressed by equation (4.6). Again equation (4.6) reduces 
to equation (4.3) if x is set equal to zero. 
If equation (4.6) were integrated at this stage of the deri- 
vation, an expression for the free surface as a function of x and 
t would be obtained. It is better to delay the integration and 
consider the possibility of obtaining some information about the 
pressure at the depth z below the free surface. 
The pressure can be found immediately from consideration of 
Somerset (oo) 4 (2665 C2875 (262), (25225 eucl (2526))5 eon 
equation (2.9), the value of © is known for z = 0. From equation 
(2.6), 6 and ®, as a function of z follow, and substitution of 
@, as a function of x, z, and t into equation (2.7) gives the 
pressure. Equations (2.9), (2.6), and (2.7) are perfectly general. 
In particular, if equation (2.25) is the free surface, then equa- 
tion (2.24) must be the potential function, and equation (2.26) 
must represent the pressure. Integration over the parameter, p , 
does not affect these relationships and the pressure as a function 
* Within the linear approximation. 
- 39 - 
